IMC 2024

Determine all pairs $(a,b) \in \mathbb{C} \times \mathbb{C}$ of complex numbers satisfying $|a|=|b|=1$ and $a+b+a\overline{b} \in \mathbb{R}$ .

For $n=1,2,\dots$ let $$S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),$$ where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$ .

For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0,1\}$ , such that $A^2$ is the matrix of all ones?

Let $g$ and $h$ be two distinct elements of a group $G$ , and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$ . Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$ . Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$ ).

Let $n>d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1,\dots,x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that the convex hull of $x_1,\dots,x_n$ contains $p$ . Prove that if $p,q \in B$ and the distance of $p$ from the origin is smaller than the distance of $q$ from the origin, then $f(p) \ge f(q)$ .

Prove that for any function $f:\mathbb{Q} \to \mathbb{Z}$ , there exist $a,b,c \in \mathbb{Q}$ such that $a<b<c$ , $f(b) \ge f(a)$ and $f(b) \ge f(c)$ .

Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and $$(A^2+B^2)(A^4+B^4)=A^5+B^5.$$ Find all possible values of $\det(AB)$ for the given $n$ .

Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$ , and the recurrence relation $$x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \text{for} n \ge 1.$$ Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies $$\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.$$

A matrix $A=(a_{ij})$ is called nice, if it has the following properties: (i) the set of all entries of $A$ is $\{1,2,\dots,2t\}$ for some integer $t$ ; (ii) the entries are non-decreasing in every row and in every column: $a_{i,j} \le a_{i,j+1}$ and $a_{i,j} \le a_{i+1,j}$ ; (iii) equal entries can appear only in the same row or the same column: if $a_{i,j}=a_{k,\ell}$ , then either $i=k$ or $j=\ell$ ; (iv) for each $s=1,2,\dots,2t-1$ , there exist $i \ne k$ and $j \ne \ell$ such that $a_{i,j}=s$ and $a_{k,\ell}=s+1$ .

Prove that for any positive integers $m$ and $n$ , the number of nice $m \times n$ matrixes is even. For example, the only two nice $2 \times 3$ matrices are $\begin{pmatrix} 1 & 1 & 12 & 2 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 & 32 & 4 & 4 \end{pmatrix}$ .

We say that a square-free positive integer $n$ is almost prime if $$n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx$$ for all integers $x$ , where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$ . Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$ ), $p$ is a prime divisor of an almost prime integer $n$ , and $p \equiv 1 \pmod{r}$ . Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$ . (An integer $n$ is called square-free if it is not divisible by $d^2$ for any integer $d>1$ .)